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Obstructions to the existence of Møller maps

Marco Benini, Alastair Grant-Stuart, Giorgio Musante, Alexander Schenkel

Abstract

Møller maps are identifications between the observables of a perturbatively interacting physical system and the observables of its underlying free (i.e. non-interacting) system. This work studies and characterizes obstructions to the existence of such identifications. The main results are existence and importantly also non-existence theorems, which in particular imply that Møller maps do not exist for non-Abelian Chern-Simons and Yang-Mills theories on globally hyperbolic Lorentzian manifolds. These results are obtained through homological algebra techniques which are of independent interest in the analysis of classical field theories.

Obstructions to the existence of Møller maps

Abstract

Møller maps are identifications between the observables of a perturbatively interacting physical system and the observables of its underlying free (i.e. non-interacting) system. This work studies and characterizes obstructions to the existence of such identifications. The main results are existence and importantly also non-existence theorems, which in particular imply that Møller maps do not exist for non-Abelian Chern-Simons and Yang-Mills theories on globally hyperbolic Lorentzian manifolds. These results are obtained through homological algebra techniques which are of independent interest in the analysis of classical field theories.
Paper Structure (6 sections, 5 theorems, 91 equations)

This paper contains 6 sections, 5 theorems, 91 equations.

Table of Contents

  1. Introduction and summary
  2. Comparison to perturbative algebraic (quantum) field theory:
  3. Notation and conventions
  4. BV formalism and L-infinity-algebras
  5. Møller maps for finite-dimensional systems
  6. Møller maps for classical field theories

Key Result

Proposition 4.2

A Møller map $K = \sum_{n=0}^\infty\lambda^n\,K_n$ exists if and only if the tower of successive obstructions given by the cohomology classes is trivial.

Theorems & Definitions (11)

  • Definition 4.1
  • Proposition 4.2
  • Theorem 4.3
  • proof
  • Theorem 4.4
  • proof
  • Definition 5.1
  • Theorem 5.2
  • proof
  • Theorem 5.3
  • ...and 1 more